The Densest Subgraph Problem with a Convex/Concave Size Function

In the densest subgraph problem, given an edge-weighted undirected graph $G=(V,E,w)$, we are asked to find $S\subseteq V$ that maximizes the density, i.e., $w(S)/|S|$, where $w(S)$ is the sum of weights of the edges in the subgraph induced by $S$. This problem has often been employed in a wide variety of graph mining applications. However, the problem has a drawback; it may happen that the obtained subset is too large or too small in comparison with the size desired in the application at hand. In this study, we address the size issue of the densest subgraph problem by generalizing the density of $S\subseteq V$. Specifically, we introduce the $f$-density of $S\subseteq V$, which is defined as $w(S)/f(|S|)$, where $f:\mathbb{Z}{\geq 0}\rightarrow \mathbb{R}{\geq 0}$ is a monotonically non-decreasing function. In the $f$-densest subgraph problem ($f$-DS), we aim to find $S\subseteq V$ that maximizes the $f$-density $w(S)/f(|S|)$. Although $f$-DS does not explicitly specify the size of the output subset of vertices, we can handle the above size issue using a convex/concave size function $f$ appropriately. For $f$-DS with convex function $f$, we propose a nearly-linear-time algorithm with a provable approximation guarantee. On the other hand, for $f$-DS with concave function $f$, we propose an LP-based exact algorithm, a flow-based $O(|V|^3)$-time exact algorithm for unweighted graphs, and a nearly-linear-time approximation algorithm.